The application of optimization to power systems has become so common that it deserves treatment as a distinct subject. The abundance of optimization problems in power systems can give the impression of diversity, but in truth most are merely layers on a common core: the steady-state description of power flow in a network. In this book, many of the most prominent examples of optimization in power systems are unified under this perspective.

As suggested by the title, this book focuses exclusively on convex frameworks, which by reputation are phenomenally powerful but often too restrictive for realistic, non-convex power system models. In Chapter 3, the application of classical and recent mathematical techniques yields a rich spectrum of convex power flow approximations ranging from high tractability and low accuracy to slightly reduced tractability and high accuracy. The remaining chapters explore problems in power system operation, planning, and economics, each consisting of details layered on top of the convex power flow approximations. Because all formulations can be solved using standard software packages, only models are presented, which is a departure from most books on power systems. It is a major perk of convex optimization that the user often does not need to program an algorithm to proceed.

I should comment that this book is not an up-to-date exposition of power system applications or optimization theory and that, inevitably, many important topics in both fields have been omitted. My intention has rather been to bridge modern convex optimization and power systems in a rigorous manner. While I have attempted to be mathematically self-contained, the pace assumes an advanced undergraduate level of mathematical exposure (linear algebra, calculus, and some probability) as well as familiarity with power systems and optimization. This book could be used in a course on power system optimization or as a mathematical supplement to a course in power system design, operation, or economics. It is my hope that it will also prove useful to researchers in power systems with an interest in optimization and vice versa, and to industry practitioners seeking firm foundations for their optimization applications.

Recent history

Streetlights, subways, the Internet, this book you are reading now – it is difficult to imagine life without such amenities, all enabled by electric power. To support such a vast set of technologies, electric power systems have grown into some of the most complex and expensive machines in existence. While much of this growth resembles an organic process more than deliberate design, the advent of computing is enabling us more and more to direct the evolution of power systems toward greater efficiency, reliability, and versatility.

At the time of writing, the complexity of power systems is poised to take off. This is largely due to shifts toward renewable energy production and the active involvement of power consumers through demand response, as well as our still-developing handle on economic deregulation. To meet these challenges, new computational tools will be developed, and the most ubiquitous computation in power systems is optimization. An objective of this book is to simplify and unify various topics in power system optimization so as to provide a firm foundation for future developments.

At the heart of most power system optimizations are the equations of the steady-state, single-phase approximation to alternating current power flow in a network. Well-known problems like optimal power flow, reconfiguration, and transmission planning all consist of details layered on top of power flow. Nodal prices, a core component of electricity markets, are obtained from the dual of optimal power flow. It is therefore most unfortunate that the power flow equations are nonconvex, making all of these optimizations extremely difficult. We are thus faced with a tradeoff between realistic models that are too hard to solve at practical scales and tractable approximations.

For many years, linear programming (LP) was the most general efficiently solvable optimization class, and so many large-scale power system models were based on linear power flow approximations or even simpler descriptions like network flow or a real power balance. At the other extreme, a number of nonlinear programming (NLP) algorithms were developed for exact, nonconvex models.

The day-to-day and long-term operation of power systems is made up of a sprawling collection of engineering and economic tasks that subsumes most of the contents of this book. This chapter does not address the full scope of power system operation but rather a handful of relevant problems that are amenable to convex optimization. As will be seen, many of these problems consist of details layered on top of the optimal power flow formulations of Chapter 3. The interested reader is referred to Wood and Wollenberg [1] for broad coverage of power system operation.

Of particular relevance to this chapter is Theorem 3.1 of Section 3.3.1, which guarantees that under certain conditions, the SOC optimal power flow relaxation is exact in radial networks. It is therefore a great boon that distribution systems, the portions of power systems that convey low-voltage electricity from substations to end users, are almost always operated radially. Until recently, distribution systems were essentially passive, predictable energy consumers. The shift from fuel-based centralized power plants to distributed and renewable generation and the new, active role of loads like electric vehicles and smart buildings are transforming distribution systems into highly actuated, potentially volatile consumers and producers of electric power. Consequently, many of the formulations in this chapter are both highly tractable and relevant when specialized to radial networks. This is especially true in Section 4.4, which deals exclusively with power flow in radial networks.

Multi-period optimal power flow

Optimal power flow routines are run every few minutes to update device and resource settings in response to the constantly changing conditions of power systems. The dynamic couplings present over these time scales justify linking these routines over successive time periods. This section formulates the multi-period optimal power flow and elucidates its application through generator ramp constraints and energy storage.

Essentially, multi-period optimal power flow is just a sequence of ordinary optimal power flow routines strung together by dynamic costs and constraints.

Semidefinite programming (SDP) is an optimization model with vector or matrix variables, where the objective to be minimized is linear, and the constraints involve affine combinations of symmetric matrices that are required to be positive (or negative) semidefinite. SDPs include as special cases LPs, QCQPs, and SOCPs; they are perhaps the most powerful class of convex optimization models with specific structure, for which efficient and well-developed numerical solution algorithms are currently available.

SDPs arise in a wide range of applications. For example, they can be used as sophisticated relaxations (approximations) of non-convex problems, such as Boolean problems with quadratic objective, or rank-constrained problems. They are useful in the context of stability analysis or, more generally, in control design for linear dynamical systems. They are also used, to mention just a few, in geometric problems, in system identification, in algebraic geometry, and in matrix completion problems under sparsity constraints.

11.1 From linear to conic models

In the late 1980s, researchers were trying to generalize linear programming. At that time, LP was known to be solvable efficiently, in time roughly cubic in the number of variables or constraints. The new interior-point methods for LP had just become available, and their excellent practical performance matched the theoretical complexity bounds. It seemed, however, that, beyond linear problems, one encountered a wall.

Optimization refers to a branch of applied mathematics concerned with the minimization or maximization of a certain function, possibly under constraints. The birth of the field can perhaps be traced back to an astronomy problem solved by the young Gauss. It matured later with advances in physics, notably mechanics, where natural phenomena were described as the result of the minimization of certain “energy” functions. Optimization has evolved towards the study and application of algorithms to solve mathematical problems on computers.

Today, the field is at the intersection of many disciplines, ranging from statistics, to dynamical systems and control, complexity theory, and algorithms. It is applied to a widening array of contexts, including machine learning and information retrieval, engineering design, economics, finance, and management. With the advent of massive data sets, optimization is now viewed as a crucial component of the nascent field of data science.

In the last two decades, there has been a renewed interest in the field of optimization and its applications. One of the most exciting developments involves a special kind of optimization, convex optimization. Convex models provide a reliable, practical platform on which to build the development of reliable problem-solving software. With the help of user-friendly software packages, modelers can now quickly develop extremely efficient code to solve a very rich library of convex problems.